SpaceX doesn't catch anything at the moment.
The first stage lands itself either at the launch site or on the autonomous barge at sea.
The fairings for the second stage get pulled out of the ocean by ship after they splash down. They used to try to catch these and that's what you're thinking of but they weren't that successful at it. I believe they redesigned the fairings to be okay spending a short amount of time in salt water and they seem to be having a pretty good success rate for this.
The second stage of the Falcon burns up in the atmosphere. Their Starship project is attempting to fix this.
The advantage of this approach for Rocket Lab is that they'll be able to re-use the first stage. At the moment SpaceX is the only one doing that and it's why they have such a commanding position in the market. I haven't seen an explanation for why (I'm sure it's out there, I just haven't seen it) but my guess is that the Electron is a much smaller rocket than the Falcon and they can't spare the lift capacity to propulsively land the vehicle.